Seminar des SFB/TRR 326 GAUS Heights of Modular Polynomials

For every positive integer N, the modular polynomial Φ_N(X,Y) has integer coefficients and vanishes precisely at pairs of j-invariants of elliptic curves linked by a cyclic isogeny of order N. These polynomials have applications in cryptography and define integral (but singular) models for the modular curves X₀(N). Their coefficients grow rapidly with N. In this talk, I will explain recent joint work with Fabien Pazuki and Desirée Gijón Gómez obtaining explicit upper and lower bounds on the size of these coefficients. Our methods also lead to explicit bounds on the heights of Hecke images. If time allows, I can also outline analogous results for Drinfeld modular polynomials.